B= [($\frac{1}{√x}$ +$\frac{1}{√y}$ ).$\frac{2}{√x + √y}$ +$\frac{1}{x}$ + $\frac{11}{y}$] : $\frac{√x^{3} + y√x +x√y + √y^{3}}{√x^{3}y + √xy^{3}}$

0 bình luận về “B= [($\frac{1}{√x}$ +$\frac{1}{√y}$ ).$\frac{2}{√x + √y}$ +$\frac{1}{x}$ + $\frac{11}{y}$] : $\frac{√x^{3} + y√x +x√y + √y^{3}}{√x^{3}y + √xy^{3}}$”

1. Đáp án:

   $\begin{array}{l}
   a)Dkxd:x > 0;y > 0\\
   B = \left[ {\left( {\dfrac{1}{{\sqrt x }} + \dfrac{1}{{\sqrt y }}} \right).\dfrac{2}{{\sqrt x  + \sqrt y }} + \dfrac{1}{x} + \dfrac{1}{y}} \right]:\\
   \dfrac{{\sqrt {{x^3}}  + y\sqrt x  + x\sqrt y  + \sqrt {{y^3}} }}{{\sqrt {{x^3}y}  + \sqrt {x{y^3}} }}\\
    = \left[ {\dfrac{{\sqrt x  + \sqrt y }}{{\sqrt {xy} }}.\dfrac{2}{{\sqrt x  + \sqrt y }} + \dfrac{{x + y}}{{xy}}} \right]:\dfrac{{x\left( {\sqrt x  + \sqrt y } \right) + y\left( {\sqrt x  + \sqrt y } \right)}}{{\sqrt {xy} \left( {x + y} \right)}}\\
    = \left( {\dfrac{2}{{\sqrt {xy} }} + \dfrac{{x + y}}{{xy}}} \right).\dfrac{{\sqrt {xy} \left( {x + y} \right)}}{{\left( {\sqrt x  + \sqrt y } \right)\left( {x + y} \right)}}\\
    = \dfrac{{2\sqrt {xy}  + x + y}}{{xy}}.\dfrac{{\sqrt {xy} }}{{\sqrt x  + \sqrt y }}\\
    = \dfrac{{{{\left( {\sqrt x  + \sqrt y } \right)}^2}}}{{\sqrt {xy} }}.\dfrac{1}{{\sqrt x  + \sqrt y }}\\
    = \dfrac{{\sqrt x  + \sqrt y }}{{\sqrt {xy} }}\\
    = \dfrac{1}{{\sqrt y }} + \dfrac{1}{{\sqrt x }}\\
   b)x.y = 16\\
    \Rightarrow y = \dfrac{{16}}{x}\\
    \Rightarrow \sqrt y  = \dfrac{4}{{\sqrt x }}\\
    \Rightarrow \dfrac{1}{{\sqrt y }} = \dfrac{{\sqrt x }}{4}\\
    \Rightarrow B = \dfrac{1}{{\sqrt y }} + \dfrac{1}{{\sqrt x }}\\
    = \dfrac{{\sqrt x }}{4} + \dfrac{1}{{\sqrt x }}\\
   Theo\,Co – si:\\
   \dfrac{{\sqrt x }}{4} + \dfrac{1}{{\sqrt x }} \ge 2.\sqrt {\dfrac{{\sqrt x }}{4}.\dfrac{1}{{\sqrt x }}}  = 2.\dfrac{1}{2} = 1\\
    \Rightarrow B \ge 1\\
    \Rightarrow GTNN:B = 1\\
   Khi:\dfrac{{\sqrt x }}{4} = \dfrac{1}{{\sqrt x }}\\
    \Rightarrow x = 4\left( {tmdk} \right)\\
    \Rightarrow y = 4
   \end{array}$

   Vậy B đạt GTNN bằng 1 khi và chỉ khi x=y=4

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